- Email: [email protected]
Mixed convection heat transfer from surface-mounted block heat sources in a horizontal channel with nanofluids
International Journal of Heat and Mass Transfer 89 (2015) 783–791
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Mixed convection heat transfer from surface-mounted block heat sources in a horizontal channel with nanofluids Mohammad Hemmat Esfe a,⇑, Ali Akbar Abbasian Arani b, Amir Hossein Niroumand b, Wei-Mon Yan c,⇑, Arash Karimipour a a b c
Department of Mechanical Engineering, Faculty of Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Isfahan, Iran Department of Mechanical Engineering, University of Kashan, Kashan, Iran Department of Energy and Refrigerating Air-Conditioning Engineering, National Taipei University of Technology, Taipei 10608, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 9 April 2015 Received in revised form 25 May 2015 Accepted 25 May 2015
Keywords: Mixed convection Horizontal channel Obstacle Nanofluid Thermophysical models
a b s t r a c t This paper presents laminar mixed convection flow of Al2O3/water nanofluids in a horizontal channel where two hot obstacles are mounted on the bottom wall. The governing equations are solved numerically using finite volume method and SIMPLER algorithm. Three thermophysical models including temperature-dependent and temperature-independent relations are selected for the study. The results are shown for a wide range of key parameters in the problem, i.e. Richardson number, Rayleigh number, and nanoparticles volume fraction. In addition, the effects of different aspect ratios of obstacles on average Nusselt number are examined. The findings elucidate that the difference between average Nusselt numbers obtained from the three sets of thermophysical models does not exceed 3%. The results also show that with increasing the nanofluid concentration from 0% to 5%, the average Nusselt number over the obstacles increases less than 10%. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction There are many strategies depending on selection of coolant, allowance for coolant to contact the device, allowance for phase change and so on. Generally, traditional means of free, forced or mixed convection are the most frequent to cool chips and channels formed with integrated circuit boards [1]. Many studies, both numerically and experimentally have been done on mixed convection heat transfer at different geometry [2–7]. Mixed convection in the entrance region of a horizontal semicircular duet was experimentally investigated for laminar water flow by Lei and Trupp [2]. With uniform heat input axially, measurements were made of axial and circumferential wall temperature variations, together with pressure drops across the heated section, to examine the buoyancy-induced secondary flow effects. Lin and Lin [3] performed an experimental study on mixed convective air flow through a bottom heated horizontal rectangular duct. The measured results indicated that the heat transfer enhancement is owing to the formation and development of a buoyancy driven secondary vortex flow. The onset of thermal instability was found to move upstream for a higher Grashof number and to be delayed ⇑ Corresponding authors. E-mail addresses: [email protected] (M. Hemmat Esfe), wmyan@ ntut.edu.tw (W.-M. Yan). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.100 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
for a larger Reynolds number. Wu and Perng [4] numerically examined mixed convective flow in a horizontal block-heated channel. The heat transfer enhancement has been accomplished by the installation of an oblique plate. The predictions disclosed that the installation of an oblique plate in cross-flow above an upstream block can effectively enhance the heat transfer performance. A numerical study was performed by Tsay [5] to study mixed convection in a horizontal duct with two heated blocks mounted on the bottom plate. The predicted results showed when a single baffle is located between the two heated blocks, the heat transfer performance of both the first and second heated blocks can be significantly promoted. Mixed convection with nanofluid in a horizontal curved tube was studied numerically by Akbarinia1and Behzadmehr [6] using three-dimensional elliptic governing equation. The predicted results indicated that nanoparticles volume fraction does not have a direct effect on the secondary flow and the skin friction coefficient. Chiu et al. [7] investigated numerically mixed convection in horizontal ducts with radiation effects. Results revealed that radiation effects have a considerable impact on the heat transfer and would reduce the thermal buoyancy effects. Besides, the development of temperature is accelerated by the radiation effects. Habchi and Acharya [8] studied the mixed convection in a vertical channel with a hot obstacle on its right wall. They found that Reynolds, Richardson, and Rayleigh numbers have an important
784
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
Nomenclature g H h k Kb M Nu P p Ra Re Ri T U, V u, v x, y w
gravitational acceleration, m s2 channel height, m component height, m thermal conductivity, kW m1 K1 Boltzmann’s constant (Kb = 1.3807 1023 J K1) molecular weight, kg/mol Nusselt number dimensionless pressure pressure, Pa Rayleigh number Reynolds number Richardson number temperature, K dimensionless velocity components components of velocity, m s1 cartesian coordinates, m component width, m
role on the rate of heat transfer. Choi et al. [9] investigated mixed convection in an inclined channel with discrete heat sources. The results showed that the inclination angle was an effective parameter on the rate of heat transfer. Another interesting research has been done by Braaten and Patankar et al. [10] in a channel with an array of hot components. In this research the effects of Buoyancy force and Prandtl number on the Nusselt number have been investigated. Incropera et al. [11] examined forced convection of cooling of electronic components. They found that the forced convection has a very important role in cooling of electronic components. Adel Hamouchi et al. [12] conducted a numerical study on mixed convection in a channel with hot protruding components. In this research the effect of Grashof and Reynolds numbers have been investigated. In a similar study, Yang et al. [13] investigated preceding research in an inclined channel. The results showed the effect of inclination angle on the rate of heat transfer and average Nusselt number. Boutina et al. [14] continued the research of Adel Hamouchi in an inclined channel. But, they studied the geometrical effects on the average Nusselt number. They found that with the increase of surface of hot components the average Nusselt number decreases. Also, they investigated the effect of inclination angle on average Nusselt number. The low thermal conductivity of fluids such as water and ethylene glycol causes to find innovative ways to enhance the thermal conductivity of mentioned fluids. One of the innovative ways to improve the heat transfer is use of nanoparticles dispersed in a base fluid, known as nanofluid. Many researchers have experimentally investigated the nanofluid properties [15,16] and have presented different models for calculating the nanofluid properties [17]. In recent years, some variable properties models have been presented by researchers. In this regard, some well-known researchers such as Khanafar, Corcione and Nguyen have presented models for various nanofluids. A good summary of results can be found in Khanafar and Vafai [18]. There are few works about mixed convection in channels that the nanofluid has been used as working fluid. Juan et al. [19] studied forced convection in a cylindrical channel using Cu–water nanofluid. Heidari et al. [20] investigated the effect of nanoparticles volume fraction on forced convection in a sinusoidal-wall channel. They concluded that heat transfer could be enhanced by the addition of nanoparticles into base fluid. Effect of Cu–water nanofluid in a rectangular horizontal channel has been studied by Santa et al. [21]. Their results show that the rate of heat transfer increased with increases in the Reynolds
Greek letters q density, kg m3 b coefficient of volume expansion, K1 a thermal diffusivity m kinetic viscosity, m2 s1 l dynamic viscosity, kg m1 s1 h dimensionless temperature u solid volume fraction Subscripts eff effective coefficient f fluid state nf nanofluid o reference state p nanoparticles s solid state
number as well as nanoparticles volume fractions. In all of the mentioned researches, the nanofluid with constant properties has been used. In the present study the problem of mixed convection of nanofluid in a channel with hot components has been investigated numerically. The Al2O3–water nanofluid with variable properties has been used as the cooling medium. 2. Analysis 2.1. Problem statement Fig. 1 shows the geometrical dimensions of the channel and position of the hot obstacles. As shown in Fig. 1, a horizontal channel with two hot obstacles has been considered as the basic model. Two hot obstacles are kept at a constant temperature while located in distances L1 and L2 from the inlet of the channel. The height and width of obstacles have been denoted with h and w, respectively. The nanofluid enters to the channel with uniform velocity, U0, and uniform temperature, T0. The top and bottom walls are kept insulated. The length of the channel after second obstacles is chosen to be sufficiently far downstream so that the fully-developed flow condition is satisfied. 2.2. Governing equations With the following dimensionless parameters,
X¼
x ; H
Ra ¼
Y¼
y ; H
U¼
bf gH3 ðT h T 0 Þ
af mf
;
u ; U0 Re ¼
V¼
v U0
U 0 qf H
lf
;
;
P¼ Pr ¼
p
qnf U 20
;
h¼
ðT T 0 Þ ðT h T 0 Þ
mf Ra ; Ri ¼ af PrRe2 ð1Þ
the governing equations for laminar, steady mixed convection including continuity, momentum and energy equations are as follows:
@U @V þ ¼0 @X @Y U
@U @U @P 1 þV ¼ þ @X @Y @X Re 1 @ @U @ @U þ lnf lnf @X @Y @Y mf qnf @X
ð2Þ
ð3Þ
785
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
Fig. 1. Schematic of the present problem.
U
@V @V @P 1 þV ¼ þ @X @Y @Y Re
1
mf qnf
@ @X
lnf
@V @X
þ
@ @Y
lnf
@V @Y
ðqbÞnf Ra h þ qnf bf PrRe2 U
Table 1 Results of grid independence examination.
ð4Þ
@h @h anf 1 1 @ @h @ @h þ knf knf þV ¼ @X @Y @X @Y @Y af knf PrRe @X
ð5Þ
In Eq. (1), density, heat capacity, thermal expansion coefficient, and thermal conductivity of nanofluid are as followings:
qnf ¼ ð1 uÞqf þ uqp
ð6Þ
ðqcp Þnf ¼ ð1 uÞðqcp Þf þ ðqcp Þp
ð7Þ
ðqbÞnf ¼ ð1 uÞðqbÞf þ uðqbÞp
ð8Þ
anf ¼
knf ðq cp Þnf
Grids
Nu of 1st obstacle
Nu of 2nd obstacle
310 51 347 61 415 71 401 81 451 81
7.4186 7.4983 7.6429 7.6999 7.7106
3.8554 3.9359 4.0950 4.1167 4.1169
Table 2 Comparison of the predicted average Nusselt number with those in Ref. [8]. Ri = Gr/Re2
Present study
Ref. [8]
Error (%)
0.1 1.0 3.0 5.0
4.4393 3.6772 3.4487 3.2533
4.4062 3.5709 3.3936 3.3621
0.7 2.9 1.6 3.3
ð9Þ
The calculation of thermal conductivity and viscosity of nanofluid will be mentioned in the next sub-section. The boundary conditions are as follows:Inlet:
U ¼ 1;
V ¼ 0;
h¼0
ð10Þ
Outlet:
@U ¼ 0; @X
V ¼ 0;
@h ¼0 @X
ð11Þ
Top wall:
U ¼ 0;
@h ¼0 @X
V ¼ 0;
ð12Þ
Bottom horizontal wall with two heat sources:
Y ¼ 0;
0X
L1 : U ¼ V ¼ 0; H
@h ¼0 @Y
On the first hot component surfaces; U ¼ V ¼ 0; Y ¼ 0;
@h ¼0 @Y
On the second hot component surfaces;
Y ¼ 0;
2.3. Nanofluid properties
h¼1
L1 þ w L2 X : U ¼ V ¼ 0; H H
U ¼ V ¼ 0;
Fig. 2. Comparison of the predicted results with those of Ref. [26].
L1 L1 þ w X : H H
L2 L2 þ w X : H H
h¼1
L2 þ w L X : U ¼ V ¼ 0; H H
@h ¼0 @Y
ð13Þ
In the present study, the thermophysical properties of the nanofluid have been calculated with two variable properties models and one constant property model. The models of Khanafar and Vafai [18] and Corcione [22] are considered as variable property models, while the MG model [23] and Brinkman model [24] are considered as constant properties models for computing the nanofluid properties. Khanafar and Vafai [18] proposed the following equations for dynamic viscosity and thermal conductivity of Al2O3/water nanofluids.
786
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
Fig. 3. Variations of the streamlines and isotherms in the channel versus Richardson number Ri at Ra = 104 for pure fluid (solid line) and nanofluid with u = 0.05 (dashed line).
Fig. 4. Variations of the streamlines and isotherms in the channel versus Rayleigh number at Ri = 0.1 for pure fluid (solid line) and nanofluid with u = 0.05 (dashed line).
lnf ¼ 0:4491 þ
0:2246 knf 1 ¼ 0:9843 þ 0:398u0:7383 dp kf
28:837 u2 þ 0:574u 0:1634u2 þ 323:053 2 T T
þ 0:0132u3 2354:735
u T
3
þ 23:498
u2 2 dp
3:0185
u3 2
dp
ð14Þ
3:9517
u T
þ 34:034
u2 T3
lnf ðTÞ lf ðTÞ
þ 32:509
u T2
!0:0235
ð15Þ
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
787
Fig. 5. Effects of Rayleigh number on average Nusselt numbers of the first and second obstacles for different Richardson numbers.
Fig. 6. Effects of volume fraction of nanoparticles on average Nusselt numbers at different Richardson numbers at Ra = 104. (a) The first obstacle; (b) the second obstacle.
where the dynamic viscosity of water is defined as
lf ðTÞ ¼ 2:414 105 10247:8=ðT140Þ
model [22], thermal conductivity and viscosity of nanofluid are given by Eqs. (17) and (18).
ð16Þ
In the above equations, dp is the nanoparticles diameter and u is the volume fraction of the nanoparticles. Also, based on Corcione
lnf 1 ¼ ; df ¼ 3:85 1010 m lf 1 34:87dp 0:3 u1:03 df
ð17Þ
788
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
Fig. 7. Effects of volume fraction of nanoparticles on average Nusselt numbers at different Rayleigh numbers and Ri = 0.1. (a) The first obstacle; (b) the second obstacle.
Fig. 8. Comparison of the predicted average Nusselt numbers using different models for thermophysical properties of the nanofluids at Ra = 104.
10 0:03 knf T kp ¼ 1 þ 4:4Re0:4 Pr0:66 u0:66 ; f T fr kf kf
Re ¼
qf kB dp pl2f dp
ð18Þ
In the above equations, Re is the nanoparticles Reynolds number, Pr is the Prandtl number of the base liquid, T is the nanofluid temperature, Tfr is the freezing point of the base liquid, kp is the
thermal conductivity of nanoparticles, kB is the Boltzmann’s constant and u is the volume fraction of the nanoparticles. As it was mentioned earlier, for comparison between the constant properties models and variable properties models, the MG [23] model for calculating the thermal conductivity of nanofluid and Brinkman model [24] for the computing viscosity of nanofluid are used. These models are presented below:
789
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
lnf ¼ lf
1
ð19Þ
ð1 uÞ2:5
knf ðkp þ 2kf Þ 2uðkf kp Þ ¼ kf ðkp þ 2kf Þ þ uðkf kp Þ
ð20Þ
2.4. Average Nusselt number The local Nusselt number Nus based on channel height (H) is calculated along the heated surface using the following definition.
Nus ¼
knf @h kf @n
ð21Þ
By integrating of local Nusselt number along the heated surface Ds, the average Nusselt number is defined as: Z sþDs 1 Nu ¼ Nus ds ð22Þ Ds s
code has been developed to solve the flow and thermal fields. To obtain desired accuracy, nonuniform grids are employed in the computational domain. The grid density is higher in the vicinity of the hot components. To check the convergence of the sequential iterative solution, the relative differences of U, V, and h at each node between two successive iterations are less than a prescribed value of 106. To demonstrate the independent of solution on the grid size, the solution has been done for various mesh sizes. The results are presented in Table 1 for Ri = 0.1 and Ra = 104. The results show that the grid system of 401 81 is fine enough to obtain accurate results. To validate the numerical procedure, laminar mixed convection in a partially blocked, vertical channel reported by Habchi and Acharya [8] is firstly solved and the predicted results are compared with those in Ref. [8]. It was found in Table 2 that the predicted results agree well with those in Ref. [8]. The maximum error of 3.3% is noted. The present work is also compared with those of Oztop and Abu-Nada [26]. In Ref. [26], the natural convection in partially heated rectangular enclosures filled with nanofluids has been investigated. As can be observed in Fig. 2, a good agreement between these results was found.
3. Numerical method The governing equations including continuity, momentum and energy equations can be solved numerically with different numerical methods. In this work, the finite volume method (FVM) is used for discretizing equations. The SIMPLER algorithm of Patankar [25] is employed to couple the velocity and pressure fields. A FORTRAN
4. Results and discussions Fig. 3 shows the effects of the Richardson number Ri on streamlines and isotherms inside the channel. Foe comparison, both the results of base fluid (solid lines) and the nanofluid with u = 0.05
10
10
(a) Ri=0.1
(b) Ri=0.1
1st component, Pure fluid 2nd component, Pure fluid
8
6
Nu
Nu
8
0.15
h/H
0.2
2 0.1
0.25
10 1st component, Pure fluid 2nd component, Pure fluid
h/H
(d) Ri=10
8
0.2
0.25
1st component, Nanofluid 2nd component, Nanofluid
8
6
Nu
Nu
0.15
10
(c) Ri=10
4
2 0.1
6
4
4
2 0.1
1st component, Nanofluid 2nd component, Nanofluid
6
4
0.15
h/H
0.2
0.25
2 0.1
0.15
h/H
0.2
Fig. 9. Effects of height of the hot obstacles on the average Nusselt number at different Richardson numbers and at Ra = 104.
0.25
790
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
(dashed lines) were simultaneously presented. The Richardson number Ri is defined as the ratio of natural convection to forced convection. At high Richardson numbers, the natural convection predominates, while at low Richardson numbers the forced convection dominates. As the flow approaches the hot components (obstacles), the streamlines are deflected towards the top wall. Therefore, in the region near the hot components, the streamlines are more densely packed. At the back of hot components, a recirculating eddy is formed. The size of recirculating eddy has an inverse relation with Ri. When the Rayleigh number is kept constant, with increases in the Richardson number the Reynolds number decreases. As a result, the thermal boundary layer increases in thickness with a decrease in Reynolds number. At low Reynolds numbers, the thermal boundary layer become thicker, the temperature gradient is more toward the normal direction of flow. In other words, at high Reynolds numbers, the temperature gradient is higher than the low Reynolds numbers. Consequently, the rate of heat transfer and average Nusselt number are more at high Reynolds numbers compared to low Reynolds numbers. The effects of the Rayleigh number on streamlines and isotherms at a fixed Richardson number are presented in Fig. 4. As it is shown in Fig. 4, with an increase in the Rayleigh number, the size of the recirculating eddy increases. At a fixed Richardson number, the Reynolds number (forced convection) increases with an increase in the Rayleigh number. Consequently, as it is expected the rate of heat transfer and Nusselt number increase.
Fig. 5 shows the effects of the Rayleigh number on the average Nusselt number of the hot obstacles in the channel. For comparison, the results of pure fluid were also presented in Fig. 5. Some important results can be achieved from Fig. 5. Firstly, at a fixed Richardson number, the average Nusselt number increases with Rayleigh number. Secondly, at a fixed Rayleigh number, the average Nusselt number decreases with Richardson number. Another point that can be found is that the average Nusselt number of second hot obstacle is less than the first hot obstacle. By moving the fluid over the first component, the mean temperature of fluid increases and consequently the temperature gradient between fluid and second obstacle would decrease. This leads to a decrease in the average Nusselt number of the second obstacle. Also, this figure indicates that the average Nusselt number of nanofluid is higher than that of pure fluid. This is because of higher thermal conductivity of the nanofluid compared to the base fluid via existence of the suspended nanoparticles. Fig. 6 indicates the effects of volume fraction of nanoparticles on the average Nusselt number of both obstacles at different Richardson numbers and the Rayleigh number of 104. As shown, the addition of nanoparticles will increase slightly the Nusselt number; especially the rate of increase is more evident at low nanofluid concentrations. The figure suggests to use nanofluids with volume fractions up to 3%, since using the nanofluids with a higher volume fraction of nanoparticles has no effect on the Nusselt number. This may be the increase of undesired viscosity
10
10
9
1st component, Pure fluid 2nd component, Pure fluid
8
8
7
7
6
6
Nu
Nu
9
(a) Ri=0.1
5
5
4
4
3
3
2
2
1
0.3
0.4
0.5
0.6
w/H
0.7
0.8
0.9
1
1
10
(c) Ri=10
1st component, Pure fluid 2nd component, Pure fluid
9
8
8
7
7
6
6
5
0.4
0.5
0.6
w/H
0.7
0.8
0.9
1
(d) Ri=10
1st component, Nanofluid 2nd component, Nanofluid
5
4
4
3
3
2
2
1
0.3
1st component, Nanofluid 2nd component, Nanofluid
10
Nu
Nu
9
(b) Ri=0.1
0.3
0.4
0.5
0.6
w/H
0.7
0.8
0.9
1
1 0.5
0.6
0.7
w/H
0.8
0.9
Fig. 10. Effects of width of the hot obstacles on the average Nusselt number at different Richardson numbers and at Ra = 104.
1
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
effects. It is also seen that the increase in Ri will result in considerable decreases of Nusselt number. Fig. 7 presents the variations of average Nusselt number versus the volume fraction of nanoparticles at different Rayleigh numbers for Ri = 0.1. It is found that at high values of Rayleigh number, the rate of changes in Nusselt number with respect to volume fraction increases. At low Rayleigh numbers in which the dominant mechanism for heat transfer is conduction, adding nanoparticles to the base fluid has little effect on the Nusselt number, however, when the convection effect increases in the fluid flow, adding nanoparticles will increase the Nusselt number in a more visible way. Fig. 8 presents a comparison between different relations used for estimation of the thermophysical properties of the nanofluids. The two variable properties models of Khanafar and Vafai [18] and Corcione [22] are compared with MG and Brinkman model. The first point that can be considered is that the variation of average Nusselt numbers for MG and Brinkman model are linear with a constant slope whereas for variable property models the slope is not constant. It can be seen that for low particle concentrations, the slope of lines is nearly the same for both two types of models. But, for higher solid volume fractions, trends are obviously different. In other words, with the increase of u, the difference becomes more significant. From Fig. 8, it is found that the behaviors of Khanafar and Vafai model and Corcione model are close together. The most important point that can be achieved is that the differences between constant property models and variable property models are not noticeable. It is due to little variation of temperature around the hot obstacles and the nature of forced convection. Consequently, for this problem, using of constant properties models does not make a noticeable error. Figs. 9 and 10 illustrate the effects of height and width of hot obstacles on the average Nusselt number at different Richardson numbers, respectively. It is obvious that the height of the obstacles affects the size of recirculation eddy. By increasing the height of the obstacles, the size of recirculating eddy increases. Therefore, the average Nusselt number decreases with an increase in the height of the obstacles. It is also observed in Fig. 10 that an increase in width of the obstacles leads to a decrease in the average Nusselt number. 5. Conclusions A numerical study has been carried out to investigate the mixed convection heat transfer in an adiabatic channel with two hot obstacles using Al2O3–water nanofluid. The effects of using different thermophysical models on the Nusselt number are investigated. The findings can be summarized as follows: The predicted average Nusselt number increases slightly with an increase in nanofluid concentration. The predicted average Nusselt numbers for both obstacles increase with a decrease in Richardson number for a fixed Rayleigh number. The effects of various thermophysical models of nanofluids on the predicted average Nusselt number are insignificant, even for high concentration of 5%. The predicted average Nusselt number decreases with an increase in height or width of the obstacles. Conflict of interest None declared.
791
Acknowledgement The financial support by the Ministry of Science and Technology, R.O.C., through the contract MOST 103-2221-E-027107-MY2 is highly appreciated. References [1] K. Sung Hoon, A Numerical Analysis of Convective Heat Transfer in Channels Simulating Electronic Components, Texas A & M University, 1993. [2] Q.M. Lei, A.C. Trupp, Experimental study of laminar mixed convection in the entrance region of a horizontal semicircular duct, Int. J. Heat Mass Transfer 34 (1991) 2361–2372. [3] W.L. Lin, T.F. Lin, Experimental study of unstable mixed convection of air in a bottom heated horizontal rectangular duct, Int. J. Heat Mass Transfer 39 (1996) 1649–1663. [4] H.W. Wu, S.W. Perng, Effect of an oblique plate on the heat transfer enhancement of mixed convection over heated blocks in a horizontal channel, Int. J. Heat Mass Transfer 42 (1999) 1217–1235. [5] Y.L. Tsay, J.C. Cheng, T.S. Chang, Enhancement on heat transfer from surfacemounted block heat sources in a duct with baffles, Numer. Heat Transfer, Part A 43 (2003) 827–841. [6] A. Akbarinia, A. Behzadmehr, Numerical study of laminar mixed convection of a nanofluid in horizontal curved tubes, Appl. Energy 27 (2007) 1327–1337. [7] H.C. Chiu, J.H. Jang, W.M. Yan, Mixed convection heat transfer in horizontal rectangular ducts with radiation effects, Int. J. Heat Mass Transfer 50 (2007) 2874–2882. [8] S. Habchi, S. Acharya, Laminar mixed convection in a partially blocked vertical channel, Int. J. Heat Mass Transfer 29 (11) (1986) 1711–1722. [9] C.Y. Choi, A. Ortega, Mixed convection in an inclined channel with a discrete heat source, Int. J. Heat Mass Transfer 36 (12) (1993) 3119–3134. [10] M.E. Braaten, S.V. Patankar, Analysis of laminar mixed convection in shrouded arrays of heated blocks, Int. J. Heat Mass Transfer 28 (9) (1985). 1699–170. [11] F.P. Incropera, Convection heat transfer in electronic equipment cooling, J. Heat Transfer 110 (4b) (1988) 1097–1111. [12] A. Hamouchi, R. Bessaih, Mixed convection air cooling of protruding heat sources mounted in a horizontal channel, Int. Commun. Heat Mass Transfer 36 (8) (2009) 841–849. [13] M. Yang, R. Yeh, J. Hwang, Mixed convective cooling of a fin in a channel, Int. J. Heat Mass Transfer 53 (4) (2010) 760–771. [14] L. Boutina, R. Bessaïh, Numerical simulation of mixed convection air cooling of electronic components mounted in an inclined channel, Appl. Therm. Eng. 31 (11–12) (2011) 2052–2062. [15] M. Hemmat Esfe, S. Saedodin, A. Asadi, A. Karimipour, Thermal conductivity and viscosity of Mg(OH)2–ethylene glycol nanofluids: Finding a critical temperature, J. Therm. Anal. Calorim. 120 (2015) 1145–1149. [16] M. Hemmat Esfe, S. Saedodin, An experimental investigation and new correlations of viscosity of ZnO-EG nanofluid at various temperatures and different solid volume fractions, Exp. Therm. Fluid Sci. 55 (2014) 1–5. [17] M. Hemmat Esfe, S. Saedodin, A. Naderi, A. Alirezaie, A. Karimipour, S. Wongwises, Modeling of thermal conductivity of ZnO-EG using experimental data and ANN methods, Int. Commun. Heat Mass Transfer 63 (2015) 35–40. [18] K. Khanafar, K. Vafai, A critical synthesis of thermophysical characteristics of nanofluids, Int. J. Heat Mass Transfer 54 (19–20) (2011) 4410–4428. [19] Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, J. Heat Transfer 125 (1) (2003) 151–155. [20] H. Heidary, M.J. Kermani, Effect of nano-particles on forced convection in sinusoidal-wall channel, Int. Commun. Heat Mass Transfer 37 (10) (2010) 1520–1527. [21] A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer due to laminar flow of copper–water nanofluid through two isothermally heated parallel plates, Int. J. Therm. Sci. 48 (2) (2009) 391–400. [22] M. Corcione, Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids, Energy Convers. Manage. 52 (2) (2011) 789–793. [23] C. Maxwell, Treatise on Electricity and Magnetism, Clarendon Press, Oxford, 1873. [24] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (4) (1952) 571–581. [25] S.V. Patankar, Numerical heat transfer and fluid flow, Hemisphere, New York, 1980. [26] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (5) (2008) 1326–1336.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Mixed convection heat transfer from surface-mounted block heat sources in a horizontal channel with nanofluids Mohammad Hemmat Esfe a,⇑, Ali Akbar Abbasian Arani b, Amir Hossein Niroumand b, Wei-Mon Yan c,⇑, Arash Karimipour a a b c
Department of Mechanical Engineering, Faculty of Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Isfahan, Iran Department of Mechanical Engineering, University of Kashan, Kashan, Iran Department of Energy and Refrigerating Air-Conditioning Engineering, National Taipei University of Technology, Taipei 10608, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 9 April 2015 Received in revised form 25 May 2015 Accepted 25 May 2015
Keywords: Mixed convection Horizontal channel Obstacle Nanofluid Thermophysical models
a b s t r a c t This paper presents laminar mixed convection flow of Al2O3/water nanofluids in a horizontal channel where two hot obstacles are mounted on the bottom wall. The governing equations are solved numerically using finite volume method and SIMPLER algorithm. Three thermophysical models including temperature-dependent and temperature-independent relations are selected for the study. The results are shown for a wide range of key parameters in the problem, i.e. Richardson number, Rayleigh number, and nanoparticles volume fraction. In addition, the effects of different aspect ratios of obstacles on average Nusselt number are examined. The findings elucidate that the difference between average Nusselt numbers obtained from the three sets of thermophysical models does not exceed 3%. The results also show that with increasing the nanofluid concentration from 0% to 5%, the average Nusselt number over the obstacles increases less than 10%. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction There are many strategies depending on selection of coolant, allowance for coolant to contact the device, allowance for phase change and so on. Generally, traditional means of free, forced or mixed convection are the most frequent to cool chips and channels formed with integrated circuit boards [1]. Many studies, both numerically and experimentally have been done on mixed convection heat transfer at different geometry [2–7]. Mixed convection in the entrance region of a horizontal semicircular duet was experimentally investigated for laminar water flow by Lei and Trupp [2]. With uniform heat input axially, measurements were made of axial and circumferential wall temperature variations, together with pressure drops across the heated section, to examine the buoyancy-induced secondary flow effects. Lin and Lin [3] performed an experimental study on mixed convective air flow through a bottom heated horizontal rectangular duct. The measured results indicated that the heat transfer enhancement is owing to the formation and development of a buoyancy driven secondary vortex flow. The onset of thermal instability was found to move upstream for a higher Grashof number and to be delayed ⇑ Corresponding authors. E-mail addresses: [email protected] (M. Hemmat Esfe), wmyan@ ntut.edu.tw (W.-M. Yan). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.100 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
for a larger Reynolds number. Wu and Perng [4] numerically examined mixed convective flow in a horizontal block-heated channel. The heat transfer enhancement has been accomplished by the installation of an oblique plate. The predictions disclosed that the installation of an oblique plate in cross-flow above an upstream block can effectively enhance the heat transfer performance. A numerical study was performed by Tsay [5] to study mixed convection in a horizontal duct with two heated blocks mounted on the bottom plate. The predicted results showed when a single baffle is located between the two heated blocks, the heat transfer performance of both the first and second heated blocks can be significantly promoted. Mixed convection with nanofluid in a horizontal curved tube was studied numerically by Akbarinia1and Behzadmehr [6] using three-dimensional elliptic governing equation. The predicted results indicated that nanoparticles volume fraction does not have a direct effect on the secondary flow and the skin friction coefficient. Chiu et al. [7] investigated numerically mixed convection in horizontal ducts with radiation effects. Results revealed that radiation effects have a considerable impact on the heat transfer and would reduce the thermal buoyancy effects. Besides, the development of temperature is accelerated by the radiation effects. Habchi and Acharya [8] studied the mixed convection in a vertical channel with a hot obstacle on its right wall. They found that Reynolds, Richardson, and Rayleigh numbers have an important
784
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
Nomenclature g H h k Kb M Nu P p Ra Re Ri T U, V u, v x, y w
gravitational acceleration, m s2 channel height, m component height, m thermal conductivity, kW m1 K1 Boltzmann’s constant (Kb = 1.3807 1023 J K1) molecular weight, kg/mol Nusselt number dimensionless pressure pressure, Pa Rayleigh number Reynolds number Richardson number temperature, K dimensionless velocity components components of velocity, m s1 cartesian coordinates, m component width, m
role on the rate of heat transfer. Choi et al. [9] investigated mixed convection in an inclined channel with discrete heat sources. The results showed that the inclination angle was an effective parameter on the rate of heat transfer. Another interesting research has been done by Braaten and Patankar et al. [10] in a channel with an array of hot components. In this research the effects of Buoyancy force and Prandtl number on the Nusselt number have been investigated. Incropera et al. [11] examined forced convection of cooling of electronic components. They found that the forced convection has a very important role in cooling of electronic components. Adel Hamouchi et al. [12] conducted a numerical study on mixed convection in a channel with hot protruding components. In this research the effect of Grashof and Reynolds numbers have been investigated. In a similar study, Yang et al. [13] investigated preceding research in an inclined channel. The results showed the effect of inclination angle on the rate of heat transfer and average Nusselt number. Boutina et al. [14] continued the research of Adel Hamouchi in an inclined channel. But, they studied the geometrical effects on the average Nusselt number. They found that with the increase of surface of hot components the average Nusselt number decreases. Also, they investigated the effect of inclination angle on average Nusselt number. The low thermal conductivity of fluids such as water and ethylene glycol causes to find innovative ways to enhance the thermal conductivity of mentioned fluids. One of the innovative ways to improve the heat transfer is use of nanoparticles dispersed in a base fluid, known as nanofluid. Many researchers have experimentally investigated the nanofluid properties [15,16] and have presented different models for calculating the nanofluid properties [17]. In recent years, some variable properties models have been presented by researchers. In this regard, some well-known researchers such as Khanafar, Corcione and Nguyen have presented models for various nanofluids. A good summary of results can be found in Khanafar and Vafai [18]. There are few works about mixed convection in channels that the nanofluid has been used as working fluid. Juan et al. [19] studied forced convection in a cylindrical channel using Cu–water nanofluid. Heidari et al. [20] investigated the effect of nanoparticles volume fraction on forced convection in a sinusoidal-wall channel. They concluded that heat transfer could be enhanced by the addition of nanoparticles into base fluid. Effect of Cu–water nanofluid in a rectangular horizontal channel has been studied by Santa et al. [21]. Their results show that the rate of heat transfer increased with increases in the Reynolds
Greek letters q density, kg m3 b coefficient of volume expansion, K1 a thermal diffusivity m kinetic viscosity, m2 s1 l dynamic viscosity, kg m1 s1 h dimensionless temperature u solid volume fraction Subscripts eff effective coefficient f fluid state nf nanofluid o reference state p nanoparticles s solid state
number as well as nanoparticles volume fractions. In all of the mentioned researches, the nanofluid with constant properties has been used. In the present study the problem of mixed convection of nanofluid in a channel with hot components has been investigated numerically. The Al2O3–water nanofluid with variable properties has been used as the cooling medium. 2. Analysis 2.1. Problem statement Fig. 1 shows the geometrical dimensions of the channel and position of the hot obstacles. As shown in Fig. 1, a horizontal channel with two hot obstacles has been considered as the basic model. Two hot obstacles are kept at a constant temperature while located in distances L1 and L2 from the inlet of the channel. The height and width of obstacles have been denoted with h and w, respectively. The nanofluid enters to the channel with uniform velocity, U0, and uniform temperature, T0. The top and bottom walls are kept insulated. The length of the channel after second obstacles is chosen to be sufficiently far downstream so that the fully-developed flow condition is satisfied. 2.2. Governing equations With the following dimensionless parameters,
X¼
x ; H
Ra ¼
Y¼
y ; H
U¼
bf gH3 ðT h T 0 Þ
af mf
;
u ; U0 Re ¼
V¼
v U0
U 0 qf H
lf
;
;
P¼ Pr ¼
p
qnf U 20
;
h¼
ðT T 0 Þ ðT h T 0 Þ
mf Ra ; Ri ¼ af PrRe2 ð1Þ
the governing equations for laminar, steady mixed convection including continuity, momentum and energy equations are as follows:
@U @V þ ¼0 @X @Y U
@U @U @P 1 þV ¼ þ @X @Y @X Re 1 @ @U @ @U þ lnf lnf @X @Y @Y mf qnf @X
ð2Þ
ð3Þ
785
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
Fig. 1. Schematic of the present problem.
U
@V @V @P 1 þV ¼ þ @X @Y @Y Re
1
mf qnf
@ @X
lnf
@V @X
þ
@ @Y
lnf
@V @Y
ðqbÞnf Ra h þ qnf bf PrRe2 U
Table 1 Results of grid independence examination.
ð4Þ
@h @h anf 1 1 @ @h @ @h þ knf knf þV ¼ @X @Y @X @Y @Y af knf PrRe @X
ð5Þ
In Eq. (1), density, heat capacity, thermal expansion coefficient, and thermal conductivity of nanofluid are as followings:
qnf ¼ ð1 uÞqf þ uqp
ð6Þ
ðqcp Þnf ¼ ð1 uÞðqcp Þf þ ðqcp Þp
ð7Þ
ðqbÞnf ¼ ð1 uÞðqbÞf þ uðqbÞp
ð8Þ
anf ¼
knf ðq cp Þnf
Grids
Nu of 1st obstacle
Nu of 2nd obstacle
310 51 347 61 415 71 401 81 451 81
7.4186 7.4983 7.6429 7.6999 7.7106
3.8554 3.9359 4.0950 4.1167 4.1169
Table 2 Comparison of the predicted average Nusselt number with those in Ref. [8]. Ri = Gr/Re2
Present study
Ref. [8]
Error (%)
0.1 1.0 3.0 5.0
4.4393 3.6772 3.4487 3.2533
4.4062 3.5709 3.3936 3.3621
0.7 2.9 1.6 3.3
ð9Þ
The calculation of thermal conductivity and viscosity of nanofluid will be mentioned in the next sub-section. The boundary conditions are as follows:Inlet:
U ¼ 1;
V ¼ 0;
h¼0
ð10Þ
Outlet:
@U ¼ 0; @X
V ¼ 0;
@h ¼0 @X
ð11Þ
Top wall:
U ¼ 0;
@h ¼0 @X
V ¼ 0;
ð12Þ
Bottom horizontal wall with two heat sources:
Y ¼ 0;
0X
L1 : U ¼ V ¼ 0; H
@h ¼0 @Y
On the first hot component surfaces; U ¼ V ¼ 0; Y ¼ 0;
@h ¼0 @Y
On the second hot component surfaces;
Y ¼ 0;
2.3. Nanofluid properties
h¼1
L1 þ w L2 X : U ¼ V ¼ 0; H H
U ¼ V ¼ 0;
Fig. 2. Comparison of the predicted results with those of Ref. [26].
L1 L1 þ w X : H H
L2 L2 þ w X : H H
h¼1
L2 þ w L X : U ¼ V ¼ 0; H H
@h ¼0 @Y
ð13Þ
In the present study, the thermophysical properties of the nanofluid have been calculated with two variable properties models and one constant property model. The models of Khanafar and Vafai [18] and Corcione [22] are considered as variable property models, while the MG model [23] and Brinkman model [24] are considered as constant properties models for computing the nanofluid properties. Khanafar and Vafai [18] proposed the following equations for dynamic viscosity and thermal conductivity of Al2O3/water nanofluids.
786
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
Fig. 3. Variations of the streamlines and isotherms in the channel versus Richardson number Ri at Ra = 104 for pure fluid (solid line) and nanofluid with u = 0.05 (dashed line).
Fig. 4. Variations of the streamlines and isotherms in the channel versus Rayleigh number at Ri = 0.1 for pure fluid (solid line) and nanofluid with u = 0.05 (dashed line).
lnf ¼ 0:4491 þ
0:2246 knf 1 ¼ 0:9843 þ 0:398u0:7383 dp kf
28:837 u2 þ 0:574u 0:1634u2 þ 323:053 2 T T
þ 0:0132u3 2354:735
u T
3
þ 23:498
u2 2 dp
3:0185
u3 2
dp
ð14Þ
3:9517
u T
þ 34:034
u2 T3
lnf ðTÞ lf ðTÞ
þ 32:509
u T2
!0:0235
ð15Þ
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
787
Fig. 5. Effects of Rayleigh number on average Nusselt numbers of the first and second obstacles for different Richardson numbers.
Fig. 6. Effects of volume fraction of nanoparticles on average Nusselt numbers at different Richardson numbers at Ra = 104. (a) The first obstacle; (b) the second obstacle.
where the dynamic viscosity of water is defined as
lf ðTÞ ¼ 2:414 105 10247:8=ðT140Þ
model [22], thermal conductivity and viscosity of nanofluid are given by Eqs. (17) and (18).
ð16Þ
In the above equations, dp is the nanoparticles diameter and u is the volume fraction of the nanoparticles. Also, based on Corcione
lnf 1 ¼ ; df ¼ 3:85 1010 m lf 1 34:87dp 0:3 u1:03 df
ð17Þ
788
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
Fig. 7. Effects of volume fraction of nanoparticles on average Nusselt numbers at different Rayleigh numbers and Ri = 0.1. (a) The first obstacle; (b) the second obstacle.
Fig. 8. Comparison of the predicted average Nusselt numbers using different models for thermophysical properties of the nanofluids at Ra = 104.
10 0:03 knf T kp ¼ 1 þ 4:4Re0:4 Pr0:66 u0:66 ; f T fr kf kf
Re ¼
qf kB dp pl2f dp
ð18Þ
In the above equations, Re is the nanoparticles Reynolds number, Pr is the Prandtl number of the base liquid, T is the nanofluid temperature, Tfr is the freezing point of the base liquid, kp is the
thermal conductivity of nanoparticles, kB is the Boltzmann’s constant and u is the volume fraction of the nanoparticles. As it was mentioned earlier, for comparison between the constant properties models and variable properties models, the MG [23] model for calculating the thermal conductivity of nanofluid and Brinkman model [24] for the computing viscosity of nanofluid are used. These models are presented below:
789
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
lnf ¼ lf
1
ð19Þ
ð1 uÞ2:5
knf ðkp þ 2kf Þ 2uðkf kp Þ ¼ kf ðkp þ 2kf Þ þ uðkf kp Þ
ð20Þ
2.4. Average Nusselt number The local Nusselt number Nus based on channel height (H) is calculated along the heated surface using the following definition.
Nus ¼
knf @h kf @n
ð21Þ
By integrating of local Nusselt number along the heated surface Ds, the average Nusselt number is defined as: Z sþDs 1 Nu ¼ Nus ds ð22Þ Ds s
code has been developed to solve the flow and thermal fields. To obtain desired accuracy, nonuniform grids are employed in the computational domain. The grid density is higher in the vicinity of the hot components. To check the convergence of the sequential iterative solution, the relative differences of U, V, and h at each node between two successive iterations are less than a prescribed value of 106. To demonstrate the independent of solution on the grid size, the solution has been done for various mesh sizes. The results are presented in Table 1 for Ri = 0.1 and Ra = 104. The results show that the grid system of 401 81 is fine enough to obtain accurate results. To validate the numerical procedure, laminar mixed convection in a partially blocked, vertical channel reported by Habchi and Acharya [8] is firstly solved and the predicted results are compared with those in Ref. [8]. It was found in Table 2 that the predicted results agree well with those in Ref. [8]. The maximum error of 3.3% is noted. The present work is also compared with those of Oztop and Abu-Nada [26]. In Ref. [26], the natural convection in partially heated rectangular enclosures filled with nanofluids has been investigated. As can be observed in Fig. 2, a good agreement between these results was found.
3. Numerical method The governing equations including continuity, momentum and energy equations can be solved numerically with different numerical methods. In this work, the finite volume method (FVM) is used for discretizing equations. The SIMPLER algorithm of Patankar [25] is employed to couple the velocity and pressure fields. A FORTRAN
4. Results and discussions Fig. 3 shows the effects of the Richardson number Ri on streamlines and isotherms inside the channel. Foe comparison, both the results of base fluid (solid lines) and the nanofluid with u = 0.05
10
10
(a) Ri=0.1
(b) Ri=0.1
1st component, Pure fluid 2nd component, Pure fluid
8
6
Nu
Nu
8
0.15
h/H
0.2
2 0.1
0.25
10 1st component, Pure fluid 2nd component, Pure fluid
h/H
(d) Ri=10
8
0.2
0.25
1st component, Nanofluid 2nd component, Nanofluid
8
6
Nu
Nu
0.15
10
(c) Ri=10
4
2 0.1
6
4
4
2 0.1
1st component, Nanofluid 2nd component, Nanofluid
6
4
0.15
h/H
0.2
0.25
2 0.1
0.15
h/H
0.2
Fig. 9. Effects of height of the hot obstacles on the average Nusselt number at different Richardson numbers and at Ra = 104.
0.25
790
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
(dashed lines) were simultaneously presented. The Richardson number Ri is defined as the ratio of natural convection to forced convection. At high Richardson numbers, the natural convection predominates, while at low Richardson numbers the forced convection dominates. As the flow approaches the hot components (obstacles), the streamlines are deflected towards the top wall. Therefore, in the region near the hot components, the streamlines are more densely packed. At the back of hot components, a recirculating eddy is formed. The size of recirculating eddy has an inverse relation with Ri. When the Rayleigh number is kept constant, with increases in the Richardson number the Reynolds number decreases. As a result, the thermal boundary layer increases in thickness with a decrease in Reynolds number. At low Reynolds numbers, the thermal boundary layer become thicker, the temperature gradient is more toward the normal direction of flow. In other words, at high Reynolds numbers, the temperature gradient is higher than the low Reynolds numbers. Consequently, the rate of heat transfer and average Nusselt number are more at high Reynolds numbers compared to low Reynolds numbers. The effects of the Rayleigh number on streamlines and isotherms at a fixed Richardson number are presented in Fig. 4. As it is shown in Fig. 4, with an increase in the Rayleigh number, the size of the recirculating eddy increases. At a fixed Richardson number, the Reynolds number (forced convection) increases with an increase in the Rayleigh number. Consequently, as it is expected the rate of heat transfer and Nusselt number increase.
Fig. 5 shows the effects of the Rayleigh number on the average Nusselt number of the hot obstacles in the channel. For comparison, the results of pure fluid were also presented in Fig. 5. Some important results can be achieved from Fig. 5. Firstly, at a fixed Richardson number, the average Nusselt number increases with Rayleigh number. Secondly, at a fixed Rayleigh number, the average Nusselt number decreases with Richardson number. Another point that can be found is that the average Nusselt number of second hot obstacle is less than the first hot obstacle. By moving the fluid over the first component, the mean temperature of fluid increases and consequently the temperature gradient between fluid and second obstacle would decrease. This leads to a decrease in the average Nusselt number of the second obstacle. Also, this figure indicates that the average Nusselt number of nanofluid is higher than that of pure fluid. This is because of higher thermal conductivity of the nanofluid compared to the base fluid via existence of the suspended nanoparticles. Fig. 6 indicates the effects of volume fraction of nanoparticles on the average Nusselt number of both obstacles at different Richardson numbers and the Rayleigh number of 104. As shown, the addition of nanoparticles will increase slightly the Nusselt number; especially the rate of increase is more evident at low nanofluid concentrations. The figure suggests to use nanofluids with volume fractions up to 3%, since using the nanofluids with a higher volume fraction of nanoparticles has no effect on the Nusselt number. This may be the increase of undesired viscosity
10
10
9
1st component, Pure fluid 2nd component, Pure fluid
8
8
7
7
6
6
Nu
Nu
9
(a) Ri=0.1
5
5
4
4
3
3
2
2
1
0.3
0.4
0.5
0.6
w/H
0.7
0.8
0.9
1
1
10
(c) Ri=10
1st component, Pure fluid 2nd component, Pure fluid
9
8
8
7
7
6
6
5
0.4
0.5
0.6
w/H
0.7
0.8
0.9
1
(d) Ri=10
1st component, Nanofluid 2nd component, Nanofluid
5
4
4
3
3
2
2
1
0.3
1st component, Nanofluid 2nd component, Nanofluid
10
Nu
Nu
9
(b) Ri=0.1
0.3
0.4
0.5
0.6
w/H
0.7
0.8
0.9
1
1 0.5
0.6
0.7
w/H
0.8
0.9
Fig. 10. Effects of width of the hot obstacles on the average Nusselt number at different Richardson numbers and at Ra = 104.
1
M. Hemmat Esfe et al. / International Journal of Heat and Mass Transfer 89 (2015) 783–791
effects. It is also seen that the increase in Ri will result in considerable decreases of Nusselt number. Fig. 7 presents the variations of average Nusselt number versus the volume fraction of nanoparticles at different Rayleigh numbers for Ri = 0.1. It is found that at high values of Rayleigh number, the rate of changes in Nusselt number with respect to volume fraction increases. At low Rayleigh numbers in which the dominant mechanism for heat transfer is conduction, adding nanoparticles to the base fluid has little effect on the Nusselt number, however, when the convection effect increases in the fluid flow, adding nanoparticles will increase the Nusselt number in a more visible way. Fig. 8 presents a comparison between different relations used for estimation of the thermophysical properties of the nanofluids. The two variable properties models of Khanafar and Vafai [18] and Corcione [22] are compared with MG and Brinkman model. The first point that can be considered is that the variation of average Nusselt numbers for MG and Brinkman model are linear with a constant slope whereas for variable property models the slope is not constant. It can be seen that for low particle concentrations, the slope of lines is nearly the same for both two types of models. But, for higher solid volume fractions, trends are obviously different. In other words, with the increase of u, the difference becomes more significant. From Fig. 8, it is found that the behaviors of Khanafar and Vafai model and Corcione model are close together. The most important point that can be achieved is that the differences between constant property models and variable property models are not noticeable. It is due to little variation of temperature around the hot obstacles and the nature of forced convection. Consequently, for this problem, using of constant properties models does not make a noticeable error. Figs. 9 and 10 illustrate the effects of height and width of hot obstacles on the average Nusselt number at different Richardson numbers, respectively. It is obvious that the height of the obstacles affects the size of recirculation eddy. By increasing the height of the obstacles, the size of recirculating eddy increases. Therefore, the average Nusselt number decreases with an increase in the height of the obstacles. It is also observed in Fig. 10 that an increase in width of the obstacles leads to a decrease in the average Nusselt number. 5. Conclusions A numerical study has been carried out to investigate the mixed convection heat transfer in an adiabatic channel with two hot obstacles using Al2O3–water nanofluid. The effects of using different thermophysical models on the Nusselt number are investigated. The findings can be summarized as follows: The predicted average Nusselt number increases slightly with an increase in nanofluid concentration. The predicted average Nusselt numbers for both obstacles increase with a decrease in Richardson number for a fixed Rayleigh number. The effects of various thermophysical models of nanofluids on the predicted average Nusselt number are insignificant, even for high concentration of 5%. The predicted average Nusselt number decreases with an increase in height or width of the obstacles. Conflict of interest None declared.
791
Acknowledgement The financial support by the Ministry of Science and Technology, R.O.C., through the contract MOST 103-2221-E-027107-MY2 is highly appreciated. References [1] K. Sung Hoon, A Numerical Analysis of Convective Heat Transfer in Channels Simulating Electronic Components, Texas A & M University, 1993. [2] Q.M. Lei, A.C. Trupp, Experimental study of laminar mixed convection in the entrance region of a horizontal semicircular duct, Int. J. Heat Mass Transfer 34 (1991) 2361–2372. [3] W.L. Lin, T.F. Lin, Experimental study of unstable mixed convection of air in a bottom heated horizontal rectangular duct, Int. J. Heat Mass Transfer 39 (1996) 1649–1663. [4] H.W. Wu, S.W. Perng, Effect of an oblique plate on the heat transfer enhancement of mixed convection over heated blocks in a horizontal channel, Int. J. Heat Mass Transfer 42 (1999) 1217–1235. [5] Y.L. Tsay, J.C. Cheng, T.S. Chang, Enhancement on heat transfer from surfacemounted block heat sources in a duct with baffles, Numer. Heat Transfer, Part A 43 (2003) 827–841. [6] A. Akbarinia, A. Behzadmehr, Numerical study of laminar mixed convection of a nanofluid in horizontal curved tubes, Appl. Energy 27 (2007) 1327–1337. [7] H.C. Chiu, J.H. Jang, W.M. Yan, Mixed convection heat transfer in horizontal rectangular ducts with radiation effects, Int. J. Heat Mass Transfer 50 (2007) 2874–2882. [8] S. Habchi, S. Acharya, Laminar mixed convection in a partially blocked vertical channel, Int. J. Heat Mass Transfer 29 (11) (1986) 1711–1722. [9] C.Y. Choi, A. Ortega, Mixed convection in an inclined channel with a discrete heat source, Int. J. Heat Mass Transfer 36 (12) (1993) 3119–3134. [10] M.E. Braaten, S.V. Patankar, Analysis of laminar mixed convection in shrouded arrays of heated blocks, Int. J. Heat Mass Transfer 28 (9) (1985). 1699–170. [11] F.P. Incropera, Convection heat transfer in electronic equipment cooling, J. Heat Transfer 110 (4b) (1988) 1097–1111. [12] A. Hamouchi, R. Bessaih, Mixed convection air cooling of protruding heat sources mounted in a horizontal channel, Int. Commun. Heat Mass Transfer 36 (8) (2009) 841–849. [13] M. Yang, R. Yeh, J. Hwang, Mixed convective cooling of a fin in a channel, Int. J. Heat Mass Transfer 53 (4) (2010) 760–771. [14] L. Boutina, R. Bessaïh, Numerical simulation of mixed convection air cooling of electronic components mounted in an inclined channel, Appl. Therm. Eng. 31 (11–12) (2011) 2052–2062. [15] M. Hemmat Esfe, S. Saedodin, A. Asadi, A. Karimipour, Thermal conductivity and viscosity of Mg(OH)2–ethylene glycol nanofluids: Finding a critical temperature, J. Therm. Anal. Calorim. 120 (2015) 1145–1149. [16] M. Hemmat Esfe, S. Saedodin, An experimental investigation and new correlations of viscosity of ZnO-EG nanofluid at various temperatures and different solid volume fractions, Exp. Therm. Fluid Sci. 55 (2014) 1–5. [17] M. Hemmat Esfe, S. Saedodin, A. Naderi, A. Alirezaie, A. Karimipour, S. Wongwises, Modeling of thermal conductivity of ZnO-EG using experimental data and ANN methods, Int. Commun. Heat Mass Transfer 63 (2015) 35–40. [18] K. Khanafar, K. Vafai, A critical synthesis of thermophysical characteristics of nanofluids, Int. J. Heat Mass Transfer 54 (19–20) (2011) 4410–4428. [19] Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, J. Heat Transfer 125 (1) (2003) 151–155. [20] H. Heidary, M.J. Kermani, Effect of nano-particles on forced convection in sinusoidal-wall channel, Int. Commun. Heat Mass Transfer 37 (10) (2010) 1520–1527. [21] A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer due to laminar flow of copper–water nanofluid through two isothermally heated parallel plates, Int. J. Therm. Sci. 48 (2) (2009) 391–400. [22] M. Corcione, Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids, Energy Convers. Manage. 52 (2) (2011) 789–793. [23] C. Maxwell, Treatise on Electricity and Magnetism, Clarendon Press, Oxford, 1873. [24] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (4) (1952) 571–581. [25] S.V. Patankar, Numerical heat transfer and fluid flow, Hemisphere, New York, 1980. [26] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (5) (2008) 1326–1336.